Abstract.

In this paper, we study the theory of linearized gravity and prove the linear stability of Schwarzschild black holes as solutions of the vacuum Einstein equations. In particular, we prove that solutions to the linearized vacuum Einstein equations centered at a Schwarzschild metric, with suitably regular initial data, remain uniformly bounded and decay to a linearized Kerr metric on the exterior region. Our method employs Hodge decomposition to split the solution into closed and co-closed portions, respectively identified with even-parity and odd-parity solutions in the physics literature. For the co-closed portion, we extend previous results of the first two authors, deriving Regge-Wheeler type equations for two gauge-invariant quantities without the earlier paper’s need of axisymmetry. For the closed portion, we build upon earlier work of Zerilli and Moncrief, wherein the authors derive a master equation for a gauge-invariant quantity in a spherical harmonic decomposition. We work with gauge-invariant quantities at the level of perturbed connection coefficients, with the initial value problem formulated on Cauchy data sets. With the choice of an appropriate gauge in each of the two portions, decay estimates on these decoupled quantities are used to establish decay of the metric coefficients of the solution, completing the proof of linear stability. Our approach complements Dafermos-Holzegel-Rodnianski’s in [6], where the linear stability of Schwarzschild is established for characteristic initial data sets.

This material is based upon work supported by the National Science
Foundation under Grant Numbers DMS 1405152 (Mu-Tao Wang). The authors would like to thank the National Center for Theoretical Sciences of National Taiwan University, where this research was initiated, for their warm hospitality. In addition, we thank Professors Simon Brendle, Sergiu Klainerman, and Ye-Kai Wang for their interests in this work.

The Schwarzschild solution of the vacuum Einstein equation in general relativity is the unique static solution that represents an isolated gravitating system of a single black hole. Studies, both theoretically and experimentally, of such a system are modeled on the Schwarzschild solution and its perturbation. The stability of the Schwarzschild solution is thus of utmost importance. More than two decades after the celebrated work of nonlinear stability of Minkowski space by Christodoulou and Klainerman [5], the nonlinear stability of Schwarzschild remains open. This paper addresses the linear stability of the Schwarzschild solution, which has a long history and rich literature involving the works of both physicists and mathematicians, and culminating in the recent breakthrough of Dafermos-Holzegel-Rodnianski [6]. This paper provides a different and simpler proof that reveals the underlying geometric structure of the vacuum Einstein equation at a more elementary level.

The question of linear stability is formulated in the following way. Consider the vacuum Einstein equation
G(g)≡Ric(g)−12R(g)g=0, of which the Schwarzschild metric g0 is solution. Let δg be a solution of the linearization of the vacuum Einstein equation at Schwarzschild:

δG|g0(δg)=0.

(1)

δg is a smooth symmetric (0,2) tensor on the background Schwarzschild spacetime. Since the Einstein equation is invariant under diffeomorphisms of the spacetime, infinitesimally any smooth co-vector field X generates another solution δg+πX of (1). Here πX is the deformation tensor of X; see Lemma 9. On the other hand, the Schwarzschild solution lies in the larger family of Kerr solutions and there are solution of (1) that correspond to Kerr perturbations. To prove the linear stability amounts to:

Firstly, formulate (1) as the initial value problem of a linear hyperbolic system of PDE’s of components of δg under a suitable gauge condition. Secondly, prove that the linearized metric coefficients of δg, after being normalized by the gauge condition, decay through a suitable foliation to a Kerr perturbation under appropriate initial conditions.

There are two main approaches to the perturbation problem:

1. Perturbation of metric coefficients. This was initiated by the seminal work of Regge-Wheeler [23]. It was shown that (1) under a suitable gauge decouples into even-parity and odd-parity perturbations, which correspond to axial and polar perturbations in Chandrasekhar [4]. Originally, Regge-Wheeler [23] discovered the master Regge-Wheeler equation in the axial or odd-parity case; more than a decade later, Zerilli [31] derived the eponymous equation in the polar or even-parity case. Later work by Moncrief [20] phrased the decoupling in terms of a gauge-invariant, connection-level quantity, which we refer to as the Zerilli-Moncrief function. See also [18, 26].

2. Perturbation in Newman-Penrose (N-P) formalism. In particular, it is known that the extreme linearized Weyl curvature components satisfy the Teukolsky equation [25], which can be further solved by separation of variables.

On the Schwarzschild background, there is a transformation theory, initiated by Wald [28] with further refinements by Aksteiner et al. [1], that relates the Regge-Wheeler equation and the Teukolsky equation. The Regge-Wheeler equation has a favorable potential which allows for a direct analysis, while this is not clear for the Teukolsky equation.

In the Schwarzschild and Kerr settings, the estimates and techniques from the study of the scalar wave equation, regarded as a “poor man’s” linearization of the vacuum Einstein equations, are expected to prove an essential ingredient in further progress on linear stability, with developments in linear stability playing a similar role in non-linear stability. A complete theory, including uniform boundedness and decay estimates, is now in place for scalar waves, first appearing in the seminal works of [7] in the Schwarzschild setting and [9] in sub-extremal Kerr, with contributions and refinements also appearing in [14, 16, 3, 10, 2, 19, 24, 17].

To get a complete theory of linear stability, one needs to prove such decay estimates for each component of a solution δg under a suitable gauge, and modulo the Kerr perturbations. Finster-Smoller [11] prove the decay estimates of Teukolsky equation on the Schwarzschild background (see also the Kerr case [12]). However, the decay estimates of the perturbed metric coefficients δg do not follow from this. Note the extreme linearized Weyl curvature components are gauge invariant quantities, while the decay of metric coefficients holds true only after a gauge condition. It seems that the reconstruction of metric coefficients from the extreme components of Weyl curvatures in the N-P formalism remains unsolved, with partial results appearing in [27]. See also [30].

The authors of [6] make use of the aforementioned transformation theory and shows that a certain second derivative of extreme linearized Weyl curvature satisfies a Regge-Wheeler type equation with a favorable potential. A double null gauge is imposed to derive that all perturbed metric coefficients decay modulo the Kerr perturbation. The initial value problem is formulated on characteristic data sets.

The current paper gives a complete theory of linear stability at the level of the metric perturbation. In the space of linearized solutions of symmetric (0,2) tensors δg, we identify the Kerr perturbation in the subspace of lower angular modes. For the higher angular modes, we work with gauge-invariant quantities at the level of perturbed connection coefficients which satisfy Regge-Wheeler type equations with favorable potentials. The Regge-Wheeler gauge and an interpolated Chandrasekhar gauge are adopted to prove the decay of all perturbed metric coefficients and complete the proof. In contrast to [6], the initial value problem is formulated on Cauchy data sets.

In order to identify the Kerr perturbation, we first decompose any smooth, symmetric (0,2) tensor δg according to angular modes

The existence part of Theorem 1 is essentially known, first appearing in the work of Zerilli [31]. See also Martel-Poisson [18]. We provide a different proof for completeness.

Theorem 2.

Under the same assumption for δg as in Theorem 1.
Assuming moreover δgℓ≥2 is compactly supported away from the bifurcation sphere on the time-slice {t=0}, there exists a smooth co-vector Xℓ≥2 such that

δgℓ≥2=πXℓ≥2+ˆδgℓ≥2,

(4)

with the gauge-normalized solution ˆδgℓ≥2 decaying pointwise through a suitable foliation.

Another decomposition (the Hodge type decomposition) of the space of symmetric (0,2) tensors is adopted to study δg. Any δg is decomposed into the closed and co-closed portions, which generalize the even-odd or axial-polar decompositions in physics literature, without any symmetry or mode assumptions. Note that K belongs to the closed part while Km,m=−1,0,1 belong to the co-closed part.

For each portion of δgℓ≥2, we decouple gauge-invariant quantities satisfying Regge-Wheeler type equations (71, 74, 97); analysis of these Regge-Wheeler type equations shows that each quantity decays to zero through a suitable foliation. The rest of the proof consists of the reconstruction of components of δgℓ≥2 from these quantities under suitable gauge choice of Xℓ≥2 and the deduction of decay of all components of δg. The choice of the gauge turned out to be quite subtle. The well-known Regge-Wheeler gauge [23] and Chandrasekhar gauge [4] both use the gauge freedom to remove four components of δgℓ≥2. However, neither seem sufficient to imply the decay of the other six components under the boundary condition. At the end, the decay of all components of δgℓ≥2 is achieved by imposing the Regge-Wheeler gauge and an interpolated Chandrasekhar gauge.

The paper is organized as follows. In Section 2, we present the Schwarzschild spacetimes as a family of static, spherically symmetric spacetimes satisfying the vacuum Einstein equations. In Section 3, we discuss linearized gravity about such spherically symmetric spacetimes. In particular, we discuss Hodge decomposition on the spheres of symmetry and decomposition into spherical harmonics. In Section 4 we present the well-known linearized Kerr family of solutions, along with the pure gauge solutions. Using such solutions, we treat the analysis of δgℓ<2 and the proof of Theorem 1 in Section 5. Subsequent sections deal with the analysis of the closed and co-closed portions of the remainder δg≥2. In Section 6, we prove decay of the co-closed portion in the Regge-Wheeler gauge, extending results from the previous [13]. In Section 7, we present the well-known Zerilli-Moncrief function as a gauge-invariant quantity, satisfying the Zerilli equation, the analysis of which is the subject of Section 8. In Section 9, we introduce the Chandrasekhar gauge, and prove decay of the closed solution under a suitable modification of the gauge. We summarize our results on δgℓ≥2 in Section 10, wherein we prove Theorem 2.

The Schwarzschild spacetimes (M,gM) comprise a family of static, spherically symmetric spacetimes, parametrized by mass M>0. Each such spacetime is vacuum; i.e., each metric gM satisfies the vacuum Einstein equations Ric(gM)=0.

The staticity and spherical symmetry of the Schwarzschild family are encoded in a number of Killing fields. In particular, we have the static Killing field, denoted T, and the rotational Killing fields, denoted Ωi, with i=1,2,3. For convenience in what follows, we collect the rotation Killing fields in the set Ω:={Ωi|i=1,2,3}. Moreover, we denote by K:={T,Ω1,Ω2,Ω3} the full set of Killing fields.

Our results concern the Schwarzschild exterior region, up to and including the future event horizon. In the course of our analysis, various coordinate systems will prove useful; we enumerate them below.

The Schwarzschild exterior, not including the event horizon, is covered by a coordinate patch (t,r,θ,ϕ) with t∈R,r>2M,(θ,ϕ)∈S2. In these standard Schwarzschild coordinates, the Schwarzschild metric has the form

gM=−(1−2Mr)dt2+(1−2Mr)−1dr2+r2˚σαβdxαdxβ,

(5)

where

˚σαβdxαdxβ:=dθ2+sin2θdϕ2

(6)

is the round metric on the unit sphere. Often we use the shorthand

μ

:=2Mr,

(7)

Δ

:=r2−2Mr.

(8)

Alternatively, we can cover this region with the Regge-Wheeler coordinates (t,r∗,θ,ϕ), with tortoise coordinate r∗ normalized as

r∗:=r+2Mln(r−2M)−3M−2Mln(M),

(9)

such that the metric

gM=−(1−μ)dt2+(1−μ)dr2∗+r2˚σαβdxαdxβ

(10)

is defined on t∈R,r∗∈R,(θ,ϕ)∈S2, with r∗=0 on the photon sphere r=3M.

A variant of the above takes t∗=t+2Mln(r−2M), with

gM=−(1−μ)dt2∗+2μdt∗dr+(1+μ)dr2+r2˚σαβdxαdxβ,

(11)

now defined for (t∗,r,θ,ϕ) coordinates satisfying t∗∈R,r>0,(θ,ϕ)∈S2. In contrast with the previous two, this coordinate system covers both the exterior region and the black hole region.

Finally, we shall refer to the double-null coordinate system (u,v,θ,ϕ), with null coordinates u and v related to the Regge-Wheeler coordinates by

u

=t−r∗,

(12)

v

=t+r∗.

(13)

With this relation, the Schwarzschild metric takes the form

gM=−(1−μ)dudv+r2˚σαβdxαdxβ.

(14)

Note that each of the above coordinate systems covers only a portion of the maximally extended Schwarzschild spacetime, globally parametrized by the Kruskal coordinates [15]. As we work only on the exterior region and event horizon, we do not employ this cumbersome coordinate system in what follows.

For more information on the Schwarzschild spacetime, we direct the reader to the comprehensive references [29, 4].

3.1. Spherically symmetric background

The analysis in this section applies to a spherically symmetric spacetime (M,g) such that the group SO(3) acts by isometry.

Let (Q,~g=∑A,B=0,1~gABdxAdxB) be a two-dimensional Lorentzian manifold with local coordinates xA,A=0,1.
Let (S2,˚σ=∑αβ=2,3˚σαβdxαdxβ) be the unit two-sphere with the standard Riemannian metric in local coordinates xα,α=2,3. We adopt the convention that any repeated index is summed. Each point on Q represents an orbit sphere, with r a positive function which represents the areal radius of each orbit sphere. We consider a general spherically symmetric spacetime in local coordinates x0,x1,x2,x3:

~gABdxAdxB+r2˚σαβdxαdxβ,A,B=0,1,α,β=2,3.

(15)

The following indices notations are adopted throughout the paper: A,B,C,⋯=0,1 for quotient indices, α,β,γ,⋯=2,3 for spherical indices, and a,b,c,⋯=0,1,2,3 for spacetime indices.

For example, the metric on the Schwarzschild spacetime with coordinates x0=t,x1=r,x2=θ,x3=ϕ has

~gABdxAdxB=−(1−2Mr)dt2+(1−2Mr)−1dr2.

(16)

The Christoffel symbols Γcab of a spherically symmetric spacetime are

ΓCAB=~ΓCAB,Γγαβ=˚Γγαβ,ΓβαA=r−1∂Ar(δβα),ΓDαβ=−r∂Dr(˚σαβ),

where ΓCAB and ˚Γγαβ are the Christoffel symbols of gAB and ˚σαβ, respectively.

For tensors defined on M, we consider two types of differential operators, ~∇A and ˚∇α. When applied to functions, ~∇A and ˚∇α are just differentiation with respect to coordinate variables xA,A=0,1 and xα,α=2,3, respectively. We then define

~∇AdxB=−~ΓBACdxC,~∇Adxα=0,˚∇αdxB=0,˚∇αdxβ=−˚Γβαγdxγ.

(17)

The operators can then be extended to associated tensor bundles, particularly the bundle of symmetric two-tensors on M. We use the notation ~□ and ˚Δ for the quotient d’Alembertian and the spherical Laplacian operators. Furthermore, we denote the volume forms for the quotient space and the unit sphere by ϵAB and ϵαβ, respectively. The operator ˚Δ can be used to characterize functions that are supported at lower angular modes.

Definition 3.

A smooth function Y on a spherically symmetric spacetime is said to be supported at ℓ<2 if

˚Δ(˚Δ+2)Y=0.

(18)

A symmetric (0,2) tensor on a general spherically symmetric spacetime is of the form

δg=hABdxAdxB+2hAαdxAdxα+hαβdxαdxβ

(19)

in our calculation.
Each component of hAB,hAα,hαβ depends on all spacetime variables.

Proposition 4.

Any symmetric (0,2) tensor δg on a spherically symmetric spacetime can be decomposed into δg=δgℓ<2+δgℓ≥2 in which the components of

δgℓ≥2=hABdxAdxB+2hAαdxAdxα+hαβdxαdxβ

are characterized by the vanishing of the integrals

∫S2hABY,∫S2(˚∇αhAα)Y,∫S2(ϵαβ˚∇αhAβ)Y,∫S2˚σαβhαβY

with respect to any function Y that is supported at ℓ<2 (Definition 3).

For more details on the proposition above, refer to the later subsection on spherical harmonic decomposition. The above discussion applies to any spherically symmetric spacetime. In the following, we collect several formulae specific to the Schwarzschild spacetime that will be used in further calculations.
The Christoffel symbols of the quotient metric (16) are

3.2. The Linearized Vacuum Einstein Equations

For an Einstein manifold (M,g) with Ric(g)=0, we consider an infinitesimal solution of the linearization of Ric(g)=0. Suppose a symmetric (0,2)-tensor hab=δgab is a linear perturbation of gab. We recall that the perturbation of the Ricci curvature δRbd satisfies

2δRbd=gae(∇a∇dheb+∇a∇bhed−∇d∇bhea−∇a∇ehbd).

(20)

On a spherically symmetric spacetime, we further decompose the last component of (19)

hαβdxαdxβ=H˚σαβdxαdxβ+^hαβdxαdxβ

into trace and traceless parts. The trace H is regarded as a function on M, while the traceless part ^hαβ is a symmetric traceless two-tensor with respect to ˚σαβ.

Expressed in this way, the linear perturbation takes the form

δg=hABdxAdxB+2hAαdxAdxα+H˚σαβdxαdxβ+^hαβdxαdxβ,

(21)

with each of hAB,hAα,H,^hαβ depending upon all spacetime variables.

Perturbing about a spherically symmetric spacetime, with radial function r, we compute the linearized Ricci tensor:

Lemma 5.

Proof.

holds when the quotient spacetime is two-dimensional. We also use ~□r2=2 on Schwarzschild and the commutation formula

˚∇γ˚∇αhAγ=˚∇α˚∇γhAγ+hAα.

(27)

∎

3.3. Hodge Decomposition

Recall that a general symmetric (0,2) tensor on a spherically symmetric spacetime is of the form

δg=hABdxAdxB+2hAαdxAdxα+(H˚σαβ+^hαβ)dxαdxβ

(28)

in our calculation.
Each component of hAB,hAα,hαβ depends on all spacetime variables. Such a decomposition is at the pointwise level. In the following, we consider a Hodge type decomposition on the spheres of symmetry.

The following Hodge type decomposition of components of δg of the form (28) follows from the appendix:

Proposition 6.

Let hAαdxAdxα be the second component of a symmetric 2-tensor δg of the form (28).
For each A=0,1, there exist functions HA and H––A on M such that

hAαdxα=[˚∇αHA+ϵβα(˚∇βH––A)]dxα.

(29)

Let ^hαβdxαdxβ be the last component a symmetric 2-tensor δg of the form (28) on M that is traceless in the
sense that ˚σαβ^hαβ=0. There exist functions H2, H––2 on M such that

Each of the decompositions above (29, 30) is invariant under the spacetime covariant derivative on Schwarzschild.

Proposition 7.

Any symmetric (0, 2) tensor δg of the form (28) can be decomposed as δg=h1+h2 where

h1=hABdxAdxB+2(˚∇αHA)dxαdxA+(H˚σαβ+˚∇α˚∇βH2−12˚σαβ˚ΔH2)dxαdxβ,

(31)

h2=2ϵβα(˚∇βH––A)dxαdxA+12(˚∇αϵγβ˚∇γH––2+˚∇βϵγα˚∇γH––2)dxαdxβ.

(32)

We note that the total number of components of δg remains ten, given by hAB,A,B=0,1, HA,A=0,1, H––A,A=0,1, H, H2, and H––2.

We refer to the portions h1 and h2 into which δg was decomposed above as the closed and co-closed portions, respectively. In the context of linearized gravity, each portion is itself a solution of (1) above. As we work within a linear theory, there is no trouble in studying, separately, the closed and co-closed pieces of a solution satisfying the linearized vacuum Einstein equations above; we simply add the two together to recover the original.

The closed and co-closed solutions generalize the even-parity (polar) and the odd-parity (axial) solutions in the physics literature, respectively. In the course of the paper, we will simply refer to them as closed and co-closed solutions.

3.4. Spherical Harmonics

The metric perturbation is comprised of objects which can be regarded as scalar, co-vector, and symmetric two-tensor quantities on the spheres of symmetry. In the course of our work, we find it necessary to decompose these objects into spherical harmonics. Owing to the Hodge decomposition of the previous subsection, wherein each of these quantities is decomposed in constituent scalar potentials, the familiar scalar spherical harmonics are seen to underly the harmonics of each object.

The scalar spherical harmonics Yℓm, indexed by integers ℓ≥0 and |m|≤ℓ, are eigenfunctions of the spherical Laplacian, with eigenvalue −ℓ(ℓ+1). That is, the Yℓm satisfy

˚ΔYℓm=−ℓ(ℓ+1)Yℓm

(33)

for ℓ≥0,|m|≤ℓ.

The eigenfunctions Yℓm form a complete, orthonormal basis of L2(S2). Note that we are normalizing

||Yℓm||L2(S2)=1.

(34)

For the co-vectors on the sphere, we have the closed harmonics

Yℓmα:=˚∇αYℓm,

(35)

and the co-closed harmonics

Xℓmα:=ϵαβ˚∇βYℓm.

(36)

Regarding symmetric two-tensors, closed harmonics have the form

Yℓmαβ:=(˚∇α˚∇β+12ℓ(ℓ+1)˚σαβ)Yℓm,

(37)

while, for the co-closed harmonics,

Xℓmαβ:=(ϵγα˚∇β+ϵγβ˚∇α)˚∇γYℓm.

(38)

We note that, for the co-vectors

˚ΔYℓmα=(1−ℓ(ℓ+1))Yℓmα,˚ΔXℓmα=(1−ℓ(ℓ+1))Xℓmα,

(39)

with support on ℓ≥1. For the symmetric-two tensors, we have

˚ΔYℓmαβ=(4−ℓ(ℓ+1))Yℓmαβ,˚ΔXℓmαβ=(4−ℓ(ℓ+1))Xℓmαβ,

(40)

with support on ℓ≥2.

Subsequently, we shall find it necessary to project the linearized solution to particular harmonics. For the closed solution h1, we use the notation

hℓmAB=~hℓmABYℓm,HℓmA=~HℓmAYℓm,Hℓm=~HℓmYℓm,Hℓm2=~Hℓm2Yℓm,

(41)

and for the co-closed solution h2,

H––ℓmA=~H––ℓmAYℓm,H––ℓm2=~H––ℓm2Yℓm,

(42)

with ~hℓmAB,~HℓmA,~H––ℓmA,~Hℓm,~Hℓm2, and ~H––ℓm2 all objects on the quotient space Q. The spherical decomposition above is done in L2 on the spheres of symmetry, so that, for example,

4.1. Linearized Kerr Solutions

Considering the Boyer-Lindquist coordinates as an extension of the standard Schwarzschild coordinates, we write the Kerr metric in the suggestive form (28)

gM,a=−(1−2Mr)dt2+(1−2Mr)−1dr2+r2˚σαβdxαdxβ−4Marsin2θdϕdt+O(a2).

(44)

We treat separately the linearized change in mass and change in angular velocity below.

Linearized Change in Mass

In the expression above, linearized mass solutions have the form

habdxadxb=δMrdt2+δMr(r−2M)2dr2,

(45)

giving infinitesimal change in mass within the Schwarzschild family. We can verify directly that the symmetric two-tensor hab satisfies the linearized vacuum Einstein equations (1). Note that linearized Schwarzschild solutions are closed solutions, supported at the lowest harmonic ℓ=0.

Linearized Change in Angular Velocity

Infinitesimal change in angular velocity appears in the linearized Kerr solution as

habdxadxb=δarϵβαY1mβdxαdt,

(46)

for m=−1,0,1. Again, direct computation shows that hab is a solution of the linearized vacuum Einstein equations (1). Such linearized Kerr solutions are co-closed solutions, supported at the harmonic ℓ=1.

Taken together, the above form the four dimensional family of linearized Kerr solutions.

Definition 8.

The Kerr solutions K,K−1,K0,K1 of the linearized vacuum Einstein equation on Schwarzschild are given by:

K=1rdt2+r(r−2M)2dr2,Km=1rϵβα˚∇βY1mdxαdt,m=−1,0,1.

4.2. Pure Gauge Solutions

First we have the following calculation lemma to identity pure gauge solutions.

Lemma 9.

Suppose G is a co-vector on Schwarzschild with G in which

G=GAdxA+(˚∇αG2)dxα+(ϵβα˚∇βG––2)dxα.

(47)

Then the deformation tensor of G decomposes into closed and co-closed parts,
πG=